There is no zombie formula. The 3 types of zombies featured in the Monster Manual aren't independently created creatures, but they are independent of each other.
The basic Zombie is one of the staple enemies of D&D, along with other swarming mooks such as Skeletons, Goblins, and Kobolds.
The Ogre Zombie is simply an interesting variant on the regular Ogre. Lower AC makes players feel more effective, with higher HP making sure the battle is no less challenging. If you look at the melee attack of the 2 creatures, you can see that it is identical, because they're effectively the same creature.
The Beholder Zombie is one of 4 types of Beholders. They all follow the same pattern, but with varying levels of power. The basic Beholder is a CR 13, with the Death Tyrant being a slightly more powerful version at CR 14, the Beholder Zombie a much weaker version at CR 5, and the Spectator bringing up the rear at CR 3.
Spectrums like this one are fairly common in the Monster Manual, particularly with iconic creatures like beholders. This allows a DM to include one of the classic monsters in their campaign regardless of whether it is a high- or low- level campaign. Other examples of this phenomenon include golems, hags, demons, devils, dragons, giants, angels, lycanthropes, and even oozes. (I could go on for longer, but I think you get the idea.)
If you want to zombify a creature, start by deciding whether you want being a zombie to make it more, less, or equally challenging. Then add zombie traits such as Undead Fortitude, lowered mental abilities, and increased Constitution. You can adjust HP and AC depending on how challenging you want it to be. You'll have to do this individually for each creature, though, if you want to maintain its unique flavour while adding zombie flavour. The DMG has instructions on how to recalculate CR once you finish modifying your creature.
The Monster Manual feats aren't limited to monsters
A feat can be taken by any creature that qualifies for the feat based on the feat's type (for example, the most common type of feat is general, but the Player's Handbook also includes the feat types item creation and metamagic) and that meets the feat's prerequisites and that fulfills any conditions explained in the feat's description, benefit, special, or note (if any).
So, while the Monster Manual feats are "typically used only by monsters" (MM 303), if a PC meets the requirements for taking such feats, nothing technically prevents the PC from taking feats from the Monster Manual.
The DM can always say No
If the DM—for whatever reason—doesn't want a feat in the campaign, the DM can make a house rule saying that feat is excluded from the campaign. As an aside, while it's usually easier to say, "No feats from this book," it's often better to ask a player's plan for his character instead then assess feat choices individually. Usually, a particular book has a mix of strong and weak feats, and banning a book means eliminating both reasonable and unreasonable choices.
Best Answer
Yes, but maybe not like you think
In Dungeons & Dragons, zombies are not contagious and being bitten by one does not turn you into a zombie. This is why you do not see anything in the Monster Manual about players being turned into zombies.
Zombies are created primarily through use of the Animate Dead spell. This spell targets "a pile of bones or a corpse of a Medium or Small Humanoid within range." Players qualify, so can be turned into Zombies by doing this. What changes is basically that the player character stops being a player character and becomes an NPC monster under the control of whomever cast the spell.
(Of course, in order to be targeted by this spell, the player character already needs to be dead.)